At the end of the first film, we see a pregnant Sarah driving through Mexico with a German Shepherd.
The Terminator (1984)
The official novelisation of the first film reveals her intentions at the time.
And so she began the Plan. When she was discharged from the hospital, she emptied her meager bank account, collected her mother's life insurance, bought the attack dog, the .357, and the jeep, then set out on the road. South. All the way to the bottom of South America, maybe. Get into the birth and raise John Connor and prepare him for the war. Where it was safe from nuclear attack.
The Terminator (novel by Randall Frakes)
In the second film, John explained that Sarah shacked up with various guys -- particularly ex-military guys -- in places like Nicaragua, in order to learn everything she could from them and pass that knowledge on to him.
JOHN: You see, we spent a lot of time in Nicaragua and places like that. For a while there, she was with this crazy ex-Green Beret guy running guns. Then there were some other guys. She'd shack up with anybody she could learn from so she could teach me how to be this great military leader. Then she gets busted. It's like, "Sorry, kid. Your mom's a psycho. Didn't you know?" It's like everything I'd been brought up to believe was all made-up bullshit. I hated her for that. But everything she said was true. She knew. And nobody believed her. Not even me.
Terminator 2: Judgment Day (1991)
John also said she was arrested for trying to blow up a "computer factory" at some point, but this was presumably after years of them living in Central/South America together.
TIM: Where'd you learn this stuff from anyway?
JOHN: From my mom. My real mom, I mean.
[...]
TIM: That her?
JOHN: Yes.
TIM: She's pretty cool, huh?
JOHN: No, she's a complete psycho. That's why she's at Pescadero. It's a mental institute, OK? She tried to blow up a computer factory, but she got shot and arrested.
Terminator 2: Judgment Day (1991)
Terminator: The Sarah Connor Chronicles exists in a timeline where only the first two films happened. According to that series, though, when John was a kid, he and Sarah spent a year and a half living in a village in Mexico...
JOHN: When I was a kid, I lived in Mexico for a year and a half... in this, uh, little village called Dejalo.
Terminator: The Sarah Connor Chronicles - S02E08 - "Mr. Ferguson is Ill Today"
... she did paramilitary training with Enrique outside of Oaxaca...
SARAH: I did paramilitary training with Enrique outside of Oaxaca. He taught John to shoot rifles.
Terminator: The Sarah Connor Chronicles - S01E02 - "Gnothi Seauton"
... and she spent three years in a "mental hospital" (likely a reference to her time in Pescadero State Hospital).
SARAH: I've had nine aliases, 23 jobs, spoken four languages and spent three years in a mental hospital for speaking the truth.
Terminator: The Sarah Connor Chronicles - S01E02 - "Gnothi Seauton"